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Two‐sided harmonic subspace extractions for the generalized eigenvalue problem
Author(s) -
Benner Peter,
Hochstenbach Michiel,
Kürschner Patrick
Publication year - 2011
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201110359
Subject(s) - eigenvalues and eigenvectors , linear subspace , mathematics , subspace topology , projection (relational algebra) , convergence (economics) , divide and conquer eigenvalue algorithm , eigendecomposition of a matrix , eigenvalue perturbation , projection method , galerkin method , mathematical analysis , pure mathematics , mathematical optimization , algorithm , finite element method , physics , dykstra's projection algorithm , economics , quantum mechanics , thermodynamics , economic growth
One crucial step of the solution of large‐scale generalized eigenvalue problems with iterative subspace methods, e.g. Arnoldi, Jacobi‐Davidson, is a projection of the original large‐scale problem onto a low dimensional subspaces. Here we investigate two‐sided methods, where approximate eigenvalues together with their right and left eigenvectors of the full‐size problem are extracted from the resulting small eigenproblem. The two‐sided Ritz‐Galerkin projection can be seen as the most basic form of this approach. It usually provides a good convergence towards the extremal eigenvalues of the spectrum. For improving the convergence towards interior eigenvalues, we investigate two approaches based on harmonic subspace extractions for the generalized eigenvalue problem. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)